| L(s) = 1 | − 2·3-s − 14·7-s + 2·9-s − 8·11-s − 24·13-s − 40·17-s + 28·21-s − 18·23-s − 18·27-s − 80·31-s + 16·33-s + 32·37-s + 48·39-s + 64·41-s − 14·43-s + 62·47-s + 98·49-s + 80·51-s + 104·53-s − 28·63-s + 162·67-s + 36·69-s − 224·71-s − 88·73-s + 112·77-s − 13·81-s + 98·83-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 2·7-s + 2/9·9-s − 0.727·11-s − 1.84·13-s − 2.35·17-s + 4/3·21-s − 0.782·23-s − 2/3·27-s − 2.58·31-s + 0.484·33-s + 0.864·37-s + 1.23·39-s + 1.56·41-s − 0.325·43-s + 1.31·47-s + 2·49-s + 1.56·51-s + 1.96·53-s − 4/9·63-s + 2.41·67-s + 0.521·69-s − 3.15·71-s − 1.20·73-s + 1.45·77-s − 0.160·81-s + 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.001629240613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001629240613\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 40 T + 800 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 62 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 526 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 32 T + 512 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 62 T + 1922 T^{2} - 62 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 104 T + 5408 T^{2} - 104 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5026 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 162 T + 13122 T^{2} - 162 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 112 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 88 T + 3872 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7298 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6238 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 88 T + 3872 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.49098178841365480738691182643, −12.11321155492525370662860545678, −11.53652881820270210764951601008, −10.84485690182995943421588084876, −10.67201670915990466508631618603, −9.831425141257696090173257446548, −9.735306376685404930078350492379, −9.062397276246976020855606469981, −8.771628310919091447810057404668, −7.56830713771649344512321824682, −7.34421757425434521149231124577, −6.83672485135516286815645436629, −6.21741548079623314595994367575, −5.71052179016484994432027106412, −5.16954715386302653092917327882, −4.24530501298743179029027947389, −3.84302256972722842021927699562, −2.56968341497803056974112147951, −2.36814428464524270892488821406, −0.02204929047879583293697831249,
0.02204929047879583293697831249, 2.36814428464524270892488821406, 2.56968341497803056974112147951, 3.84302256972722842021927699562, 4.24530501298743179029027947389, 5.16954715386302653092917327882, 5.71052179016484994432027106412, 6.21741548079623314595994367575, 6.83672485135516286815645436629, 7.34421757425434521149231124577, 7.56830713771649344512321824682, 8.771628310919091447810057404668, 9.062397276246976020855606469981, 9.735306376685404930078350492379, 9.831425141257696090173257446548, 10.67201670915990466508631618603, 10.84485690182995943421588084876, 11.53652881820270210764951601008, 12.11321155492525370662860545678, 12.49098178841365480738691182643
